# Functional Dependence between the Hamiltonian and the Modular Operator Associated with a Faithful Invariant State of a <i>W</i>*-Dynamical System

### J. de Cannière

University of California, Berkeley, USA

## Abstract

Let H and A be the hamiltonian, resp. the modular operator associated with an invariant faithful normal state ω of a *W**-dynamical system (*A*, *α*). Then *Δ = f*(*h*) for some decreasing function *f* if and only if (roughly speaking) ω is 2-passive with respect to α. It follows that under certain conditions a 3-passive state is an equilibrium (i.e. KMS) state.

## Cite this article

J. de Cannière, Functional Dependence between the Hamiltonian and the Modular Operator Associated with a Faithful Invariant State of a <i>W</i>*-Dynamical System. Publ. Res. Inst. Math. Sci. 20 (1984), no. 1, pp. 79–96

DOI 10.2977/PRIMS/1195181829